Percentage, profit and averages: one idea in three disguises

Percentage, profit and loss, and averages look like three topics, but in a bank or railway prelims they are one skill asked three ways. Each is really a statement about a ratio, and once you read them that way the long calculations the setter is hoping for collapse into a single multiplication. This is the small set of habits that does that.

A percentage is a multiplier, not a two-step sum

The slow way to add 20% to a number is to find 20% and then add it. The fast way is to multiply by 1.2 once. Every percentage change is a multiplier:

• +20% is ×1.2, −20% is ×0.8, +7% is ×1.07, and so on.
• To reverse a change you divide by the multiplier. If a price after a 25% rise is 500, the original is 500 ÷ 1.25 = 400, not 500 − 25%.

Learning the common fraction equivalents makes this instant, because a percentage of a number is easiest as a fraction: 12.5% is 1/8, 16.66% is 1/6, 20% is 1/5, 25% is 1/4, 33.33% is 1/3, 37.5% is 3/8, 62.5% is 5/8. When a question says “one number is 37.5% more than another”, read it as a 8 : 11 ratio and you have skipped the calculation entirely.

Successive change: add, then correct

When two percentage changes act one after another — a rise then a fall, two years of growth, a price marked up then discounted — the net change is a + b + ab/100, where a and b carry their own signs.

• A 20% rise followed by a 20% fall is 20 − 20 − 400/100 = −4%, a net loss, which is why “up then down by the same percent” never returns to the start.
• A 10% and 10% rise compounds to 21%, the same effective-rate idea used in compound interest.

This one formula covers population growth, depreciation, and successive discounts, so it is worth more than any of them individually.

Profit and loss: fix the base first

The whole topic turns on which price the percentage sits on.

• Profit and loss are percentages of the cost price. Selling price = cost price × (1 ± profit fraction). A 25% profit means SP = CP × 1.25.
• Discount is a percentage of the marked price. Selling price = marked price × (1 − discount fraction).
• A question that gives a marked price, a discount, and a profit is asking you to move from marked price down to selling price, then relate selling price back to cost price. Set the cost price as 100 (or as the fraction-friendly number) and let the multipliers do the work.

The favourite trap is a percentage applied to the wrong base — profit worked on the selling price, or discount on the cost. Decide the base before you write a single figure.

Averages: work with the total, never the average

An average is just total ÷ count, so the useful move is almost always to convert back to the total.

• Change in total = change in average × number of items. If the average age of 11 players rises by 2 years when a new player joins, the total rose by 2 × 11 = 22, which tells you the newcomer’s age relative to the mean directly.
• Replacement. When one value is swapped for another, the total changes by exactly the difference between them, and the average changes by that difference over the count.
• Weighted average. Two groups of sizes m and n with averages A and B have a combined average of (mA + nB)/(m + n) — the same alligation idea used for mixtures, so learning it here pays off twice.

Chasing the average figure directly is where candidates lose time and drop terms. The total is the quantity that actually conserves, so reason with it.

A worked reading

Suppose an article is marked 40% above cost and then sold at a 25% discount. Instead of picking numbers and grinding, set cost = 100. Marked price = 140. Selling price = 140 × 0.75 = 105. So the profit is 5 on 100 — a 5% profit, read off in two multiplications. The setter budgeted a paragraph of working for a question that is three moves once cost is the base of 100.

Where the marks leak

Adding percentages of different bases. 20% of one number plus 20% of another is not 20% of anything unless the numbers are equal. Keep each percentage tied to its own base.

Forgetting the ab/100 term. Two successive changes are never just a + b; the correction term is the whole point of the question.

Confusing profit on cost with margin on sale. They are different numbers; read which one the question wants.

Reasoning with the average instead of the total. Convert to the total, change it, convert back. It is one extra line that removes the error entirely.

Practice that transfers

These three read as one topic once every percentage is a multiplier, so drill them together rather than in silos. Work through the percentage, profit and averages practice set, and keep the underlying multiplication fast with calculation speed so that a 37.5% is something you see as 3/8 rather than something you compute.